The federal government officially bumped up against its borrowing limit Monday, an unhappy milestone that signals the beginning of a two-month sprint during which Congress and the White House will try to agree on whether to raise the debt ceiling while imposing spending restraints.

In the short term, hitting the debt limit means little - the Treasury Department has tools it can use to keep government running at full tilt through early August. But if the $14.294 trillion ceiling is not raised by then, the government will have to stop paying more than a third of its annual spending, which could halt workers’ salaries, contracts, entitlement benefits, interest payments or some combination of those.

“We need to have a vote to lift the debt ceiling because the consequences of not doing so would be quite serious, indeed,” White House press secretary Jay Carney told reporters traveling with President Obama. “Those who suggest otherwise are whistling past the graveyard.”

Even many Republicans say they want to raise the borrowing limit, but there the agreement ends.

The White House has called for a “clean” increase. Officials have said that they don’t want to spook the financial markets by suggesting there are conditions to raising the ceiling. But Republicans have said that is impossible and irresponsible because it would allow the government to continue piling up record annual deficits.

The GOP’s leaders are insisting on binding changes in the short term to discretionary spending, and in the long term to entitlement spending.

“Americans understand we simply can’t keep spending money we don’t have,” said House Speaker John A. Boehner, Ohio Republican. “There will be no debt limit increase without serious budget reforms and significant spending cuts - cuts that are greater than any increase in the debt limit.”

Even the White House mixed its messages. Spokesmen have insisted that conditions won’t be attached, but Mr. Obama has said he expects to have to agree to some GOP demands.

The president is also the victim of his own record. He voted against raising the debt ceiling while he was in the Senate. His advisers have since said that was a mistake.

Treasury Secretary Timothy F. Geithner alerted Congress in a letter Monday that the ceiling had been hit. He said he will temporarily stop paying into some government retirement funds, delaying the crossing of the statutory debt limit until Aug. 2.

After that, the government will be able to spend only what it gets in revenue. Since the government borrows about 40 cents of every dollar it spends, that would mean immediately halting about 40 percent of all spending.

Mr. Geithner said it could have “catastrophic economic consequences for citizens” if Congress doesn’t act.

The staggering total dollar amount of the debt is difficult to grasp, so Rep. Jeff Flake, Arizona Republican and a longtime deficit hawk, regularly releases statements measuring the size of the debt based on everyday items.

Last month, he said that if Prince William’s royal wedding to Kate Middleton cost $49 million, the U.S. debt could fund 285,714 of them. On Monday, he said the debt could buy 2.86 trillion $5 bags of Entenmann’s Pop’ems doughnut holes.

The last time the debt limit was raised was Feb. 12, 2010, when it increased by $2.19 trillion.

Democrats and Republicans have used parliamentary techniques to shield House lawmakers from having to vote on increasing the debt, but Mr. Boehner promised House Republicans that they would have a chance to have a specific vote, and that decision has given him more power and more responsibility in negotiations.

Much of the debate over spending has centered on how the U.S. got into this situation.

During the Clinton administration, the debt grew from $4.188 trillion to $5.728 trillion. The amount increased dramatically under the Bush administration, to $10.627 trillion. Since Mr. Obama became president, the amount of debt has hit $14.308 trillion. The total is more than the debt limit because not all debt is counted against the statutory ceiling.

A Congressional Budget Office report last week looked at the shift in the government’s fiscal stance in 2001 versus today and found that new spending and a poor economy were responsible for most of the deterioration, followed by Mr. Bush’s tax cuts and then Mr. Obama’s tax cuts.

Mr. Obama has convened a group of top-level lawmakers and Vice President Joseph R. Biden and asked them to work on a deal to start bringing down deficits. Mr. Obama insists that tax increases should be part of the solution, while Republicans have said entitlement cuts need to be considered instead.

On Monday, House Minority Leader Nancy Pelosi told CNBC that Medicare changes should be “on the table,” though she told Bloomberg that doesn’t mean agreeing to House Republicans’ plan to convert the program into a voucher-style system.

The issue will continue to roil the political debate for the next two months, but on Monday a group representing federal employees said the fact that their retirement funds are being tapped to supply the government with cash punctures the belief by some that government workers’ costs are overwhelming the system.

“It is clear that federal employee retirement funds aren’t bankrupt or bankrupting America. Rather, the funds are strong enough to save the nation from bankruptcy,” said Joseph A. Beaudoin, president of the National Active and Retired Federal Employees Association.